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有理点
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  rational points
     THe Nurnber of Rational points of Algebraic Curves over Finite Fields
     有限域上代数曲线有理点个数
短句来源
     The key to finding a useful elliptic curve for Elliptic Curve Cryptosystem (ECC) is to compute the order of elliptic curve rational points group over a finite field.
     寻找对椭圆曲线公钥体制(ECC)有用的椭圆曲线,关键在于求有限域上椭圆曲线有理点群的阶。
短句来源
     Explicit Classification for Torsion Subgroups of Rational Points of Elliptic Curves
     椭圆曲线有理点扭子群的明显分类
短句来源
     In this paper we show that the set of rational points of elliptic curves over finite fields of characteristic 3 is equivalent to the set of reduced principal ideals of real quadratic function field of genus 1. Therefore their discrete loga- rithms problems are equivalent.
     论文给出了特征等于3的有限域上,椭圆曲线的有理点群(除去这个点本身)与实二次函数域上的既约主理想之间的一一对应,从而使二者的离散对数问题(DLP)等价。
短句来源
     Methods Using some facts of rational points of elliptic curves and congruent numbers.
     方法利用椭圆曲线上的有理点以及同余数的相关信息。
短句来源
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  rational point
     Probenius endomorphism was first applied to speed up rational point scalar multiplication on an elliptic curve by Koblitz [53].
     Koblitz[53]最先利用Probenius自同态来快速计算椭圆曲线上的有理点标量乘。
短句来源
     In [1], Andreas Stein explicitly gave the one-to-one correspondence between the group generated by a rational point of a elliptic curve (without the point itself) over finite fields in characteristic p (p is not 2,3) and the set of reduced principal ideals of the corresponding real quadratic function fields, then their discrete logarithms problems are equivalent.
     在特征不等于2,3的情形,Andreas Stein[1]建立了有限域上椭圆曲线由一个有理点生成的群(除去这个点本身)与对应实二次函数域上的既约主理想之间的一一对应,证明了这二者的离散对数问题等价。
短句来源
     The divisor class group, often called Jacobian group, based on which hyperelliptic curve cryptosystems are constructed, is much complicated than the elliptic curve rational point group.
     超椭圆曲线密码体制所基于的除子类群,又称Jacobian群,其结构与运算比有理点群要复杂得多。
短句来源
     In [71], Miiller computed rational point scalar multiplications on elliptic curves over small finite fields of characteristic 2 by representing the multipliers into Probenius expansion, that is, r-adic expansion.
     在[71]中,M(?) ller将有理点标量乘的乘子作Frobenius展开,即将乘子表成τ-展开式,然后利用该展开式来计算特征为2的小基域上的椭圆曲线上有理点的标量乘。
短句来源
     These results, together with the recent results of K. Ono for the non-cyclic Eaors (Q), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.
     这些结果,连同新近Ono在Etors(Q)非循环情形 的结果,完全解决了E含2阶有理点时的分类和参数化问题.
短句来源
  “有理点”译为未确定词的双语例句
     In this paper, We count out the number of F_p—points on the elliptic curve E:y~2=x~3+ax(a∈F_P~*) defined over F_p by using the Jacobsthal Sum, and discuss the applications of Jacobsthal Sum in elliptic curve cryptography.
     本文利用Jacobsthal和求出了有限域F_p上椭园曲线E:y~2=X~3+ax(a∈F_p)的有理点个数,讨论了Jacobsthal和在椭园曲线密码体制中的应用。
短句来源
     Later, Koblitz's method was modified by Meier and Staffelbach[60], they proposed a method which is three times faster than Binary method for anomalous elliptic curves in characteristic 2 , and in 1997 it was improved by Solinas [104] for a class of elliptic curves in characteristic 2. Based on Koblitz's idea, Fangguo Zhang and etc.
     其后Meier与Staffelbach[60]利用Koblitz的算法思想,针对特征为2的域上的一类反常椭圆曲线,提出了一种比二元法快三倍的有理点标量乘算法。 1997年Solinas[104]又针对特征为2的域上的一类椭圆曲线改进了他们的算法。
短句来源
     Let X be a smooth and absolutely irreducible projective curve,x(F_q)the set ofrational points of X over the field F_q,and g(X)the genus of X. Let serre has proved that This paper proves that and% by constrncting the infinite Hilbert class field towers.
     设X是有限域上光滑、绝对不可约的射影曲线,表示其有理点集,g(X)表示X的亏格,令证明了本文利用构造无Hilbert类域塔方法证明了
短句来源
     By understanding the structure of the set of continuous points of real function,we prove that there is no such real function defined on the set of R n which is continuous on theset of Q n ,but non continuous on the set of ( Q n ).
     在弄清实函数连续点集结构的基础上,证明了不存在定义在Rn上的实函数f,使得f在有理点集Qn上连续,而在无理点集(Qn)上间断.
短句来源
     Implementing various cryptosystems,based on the Abel group composed of the points on elliptic curves,has already developed into an extremely important subject in cryptology research.
     以椭圆曲线上的有理点构成的Abel群为背景结构,实现各种密码体制已是公钥密码学领域的一个重要课题。
短句来源
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  rational points
We also describe the closure of orbits and then give applications to the repartition of rational points on K3 surfaces.
      
Let q be a power of p and let G(q) be the finite group of Fq-rational points of G.
      
Rational approximations of some functions at rational points
      
Construction of functions from their values at the binary-rational points of an m-dimensional cube
      
Some corollaries concerning the algebraic independence of the values of Euler's beta and gamma functions at rational points are derived.
      
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  rational point
We show the following: if there exists a k-variety which is birational to X and which has a smooth k-rational point, then X also has a k-rational point (Theorem 5.7).
      
By using Padé approximations of the first kind, a lower bound for the modulus of a linear form with integer coefficients in the values of certain hypergeometric functions at a rational point are obtained.
      
In this paper, we use Hermite-Padé approximations of the second kind to obtain a lower bound for the absolute value of a linear form, with integer coefficients, in values of polylogarithmic functions at a rational point.
      
In this paper, we use Hermite-Padé approximations of the second kind to obtain a lower bound for the absolute value of a linear form, with integer coefficients, in values of polylogarithmic functions at a rational point.
      
These results, together with the recent results of Ono for the non-cyclic torsion groups, have completely solved the problem of the explicit classification withE being a rational point of order 2.
      
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A pair of compasses is called rusty,if we can only draw the unit circleswith it.What can we do with a rusty pairof compasses?The answers give a feelingof suprising to us!From two points A and B,we candraw the points of k-section of AB,andcan draw all vertices of a regular n-gonwhich has a side AB,here n=3,4,5,6,8,12,17,257,…,etc.In fact,we can do more things:Iftake point A as origin(0,0)and B as(1,0),then we can draw all the points(x,y),here X and Y belong to some regular 2~m-extension of the rational field.

用一把两脚开度不能变化的“生锈圆规”,能不能找出连接任意两点A、B 的线段的中点来?答案是:能!这个消息将会使提出这一趣味难题的美国几何学家佩多教授感到高兴和意外.用生锈圆规所能干的事,远比人们原来想象的要多:用它能找出直线上所有的有理点,甚至所有整系数二次方程的根;用普通圆规可以作出的正多边形,用它也能作出.也许更使佩多教授感到意外的是,这些结论,并非由专业数学工作者所获得.我国一位正在自学的普通待业青年,作出了这个也许是近百年来在尺规限制作图方面最引人注目的贡献.

We give a uniform construction of the simple algebraic groups definedover a p-adic field k having the various k-indices and give a uniform construction ofthe simple groups which are the groups of the k-rational points of the simple algebraicgroups defined over k mentioned above.Then we give a uniform proof of existence forsimple algebraic groups defined over k for all k-indices in this note.

设k为p-adic域,给出了所有定义在k上,具有不同类型k-指标的单线性代数群的统一构造方法,从而给出了定义在k上单代数群的存在性的统一证明,并且给出了所有定义在k上单线性代数群的k-有理点群的统一构造。

In this paper, We count out the number of F_p—points on the elliptic curve E:y~2=x~3+ax(a∈F_P~*) defined over F_p by using the Jacobsthal Sum, and discuss the applications of Jacobsthal Sum in elliptic curve cryptography.

本文利用Jacobsthal和求出了有限域F_p上椭园曲线E:y~2=X~3+ax(a∈F_p)的有理点个数,讨论了Jacobsthal和在椭园曲线密码体制中的应用。

 
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